Satisfiable

Such a formula is indeed satisfiable if and only if at least one of its conjunctions is satisfiable, and a conjunction is satisfiable if and only if it does not contain both x and NOT x for some variable x. This can be checked in linear time.

However, suppose that for every formula φ there is some formula ψ taken from a more restricted class of formulas C, such that " is either refutable or satisfiable" → " is either refutable or satisfiable".

A formula is said to be satisfiable if it can be made TRUE by assigning appropriate logical values (i.e. TRUE, FALSE) to its variables.

An initial problem is solved by reducing it to a satisfiable conjunction of constraints.

If every formula in R of degree k is either refutable or satisfiable, then so is every formula in R of degree k+1.

If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬φ is not satisfiable in any structure and therefore refutable; then ¬¬φ is provable and then so is φ, thus Theorem 1 holds.

If this is the case, the formula is called satisfiable.

Let us call the class of all such formulas R. We are faced with proving that every formula in R is either refutable or satisfiable.

Note that the tautology problem for positive Boolean formulae remains co-NP complete, even though the satisfiability problem is trivial, as every positive Boolean formula is satisfiable.

Now is a formula of degree k and therefore by assumption either refutable or satisfiable.

Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal.

So φ is satisfiable, and we are done.

The Boolean satisfiability problem (SAT) is, given a formula, to check whether it is satisfiable.

Then, once this claim (expressed in the previous sentence) is proved, it will suffice to prove " is either refutable or satisfiable" only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ (i.

Therefore, model theorists often use "consistent" as a synonym for "satisfiable".

We have proved that φ is either satisfiable or refutable, and this concludes the proof of the Lemma.

We immediately restate it in a form more convenient for our purposes: Theorem 2. Every formula is either refutable or satisfiable in some structure.